Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place.)
a = 27,    b = 12,    ∠A = 111°

We are given the following information about a triangle:

- \( a = 27 \) (side opposite angle \( A \)),
- \( b = 12 \) (side opposite angle \( B \)),
- \( \angle A = 111^\circ \) (angle at vertex \( A \)).

To solve for the unknowns \( \angle B \), \( \angle C \), and \( c \) (the side opposite angle \( C \)), we will use the **Law of Sines**, which states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

### Step 1: Use the Law of Sines to solve for \( \sin B \)

Using the ratio between sides \( a \) and \( b \), we can write:

\[
\frac{a}{\sin A} = \frac{b}{\sin B}
\]

Substituting the known values:

\[
\frac{27}{\sin 111^\circ} = \frac{12}{\sin B}
\]

Now, solve for \( \sin B \):

\[
\sin B = \frac{12 \cdot \sin 111^\circ}{27}
\]

First, calculate \( \sin 111^\circ \), and then compute \( \sin B \).

### Step 2: Solve for angle \( B \)

Once \( \sin B \) is found, use the inverse sine function to determine the angle \( B \). There may be two possible values for \( B \) since \( \sin B \) could correspond to two different angles.

### Step 3: Solve for angle \( C \)

Use the triangle sum property \( \angle A + \angle B + \angle C = 180^\circ \) to find \( \angle C \) once \( \angle B \) is known.

### Step 4: Use the Law of Sines to solve for side \( c \)

Once \( \angle C \) is found, use the Law of Sines again to find side \( c \):

\[
\frac{a}{\sin A} = \frac{c}{\sin C}
\]

Let me calculate these steps for you.### Solution:

We found the following two possible triangles:

1. **First Triangle:**
   - \( \angle B_1 = 24.5^\circ \)
   - \( \angle C_1 = 44.5^\circ \)
   - Side \( c_1 = 20.3 \)

2. **Second Triangle:**
   - \( \angle B_2 = 155.5^\circ \)
   - \( \angle C_2 = -86.5^\circ \) (which is impossible, so this triangle doesn't exist).

Thus, only **one triangle** is possible with:
- \( \angle B = 24.5^\circ \),
- \( \angle C = 44.5^\circ \),
- \( c = 20.3 \).

Would you like further details or have any questions?

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Here are 5 related questions:
1. How does the Law of Sines work for non-obtuse triangles?
2. What if two sides and an angle not opposite either side are given?
3. How would the solution change if \( \angle A \) was acute?
4. Can you solve for a triangle with all angles but no sides given?
5. How does the Law of Cosines differ from the Law of Sines?

**Tip:** Always check the range of sine values when solving trigonometric equations, as they must lie between -1 and 1.

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions. (If an answer does not exist, enter DNE. Round your answers to one decimal place.) a = 27, b = 12, ∠A = 111°

Solve a triangle using the Law of Sines where side a = 27, side b = 12, and angle A = 111°. This detailed step-by-step guide explains how to find all possible triangles and addresses the trigonometric concepts involved. Suitable for Grades 9-12.

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